Topological classification of the 4-manifolds bridges computation theory and physics. A proof of the undecidability of the homeomorphy problem for 4-manifolds is outlined here in a clarifying way. It is shown that an arbitrary Turing machine with an arbitrary input can be encoded into the topology of a 4-manifold, such that the 4-manifold is homeomorphic to a certain other 4-manifold if and onl...

This paper explores and clarifies several issues surrounding Zeno machines and the issue of running a Turing machine for infinite time. Without a minimum hypothetical bound on physical conditions, any magical machine can be created, and therefore, a thesis on the bound is formulated. This paper then proves that the halting problem algorithm for every Turing-recognizable program and every input ...

This paper shows that a constraint on universal Turing machines is necessary for Legg's and Hutter's formal measure of intelligence to be unbiased. Their measure, defined in terms of Turing machines, is adapted to finite state machines. A No Free Lunch result is proved for the finite version of the measure.

By the Church±Turing Thesis a numerical function is computable by a physical device if and only if it is computable by a Turing machine. The `if'-part is plausible since every (suf®ciently small) Turing machine can be simulated by a computer program which operates correctly as long as suf®cient time and storage are available and no errors occur. On the other hand, every program for a modern dig...

The halting problem for Turing machines is decidable on a set of asymptotic probability one. The proof is sensitive to the particular computational model. The halting problem for Turing machines is perhaps the canonical undecidable set. Nevertheless, we prove that there is an algorithm deciding almost all instances of it. The halting problem is therefore among the growing collection of those ex...

There has been many philosophical discussions about the realizability of machines that are as intelligent as (or more) humans. Turing Test defines a certain methodology to test if a machine is intelligent or not. The machine may or may not have the same processing method as humans. Though there has always been a debate regarding the validity of the test (John Searle et al), satisfying Turing te...

Four universal Turing machines are given with strong limitations on the number of left instructions in their program. One of the machines is morever non-erasing, i.e. it does not erase any 1 written on the tape. c © 2001 Elsevier Science B.V. All rights reserved.

1. The Physical Church-Turing Thesis. Physicists often interpret the Church-Turing Thesis as saying something about the scope and limitations of physical computing machines. Although this was not the intention of Church or Turing, the Physical Church Turing thesis is interesting in its own right. Consider, for example, Wolfram's formulation: One can expect in fact that universal computers are a...

A Turing machine can be viewed as a simple kind of computer, whose operations are constrainted to reading and writing symbols on a tape, or moving along the tape to the left or right. In theory, one has proven that the computability of Turing machines is equivalent to recursive functions. This article defines and verifies the Turing machines of summation and three primitive functions which are ...

The recent advances in computer animation, motion image processing, robotics and so on prompted us to analyze computational complexity of four-dimensional pattern processing. Thus, the research of four-dimensional automata as a computational model of four-dimensional pattern processing has also been meaningful. From this viewpoint, we introduced a four-dimensional alternating Turing machine (4-...